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Mirrors > Home > ILE Home > Th. List > neg1ap0 | GIF version |
Description: -1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.) |
Ref | Expression |
---|---|
neg1ap0 | ⊢ -1 # 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1ap0 8345 | . 2 ⊢ 1 # 0 | |
2 | ax-1cn 7706 | . . 3 ⊢ 1 ∈ ℂ | |
3 | negap0 8385 | . . 3 ⊢ (1 ∈ ℂ → (1 # 0 ↔ -1 # 0)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (1 # 0 ↔ -1 # 0) |
5 | 1, 4 | mpbi 144 | 1 ⊢ -1 # 0 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 1480 class class class wbr 3924 ℂcc 7611 0cc0 7613 1c1 7614 -cneg 7927 # cap 8336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-ltxr 7798 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 |
This theorem is referenced by: m1expcl2 10308 m1expeven 10333 m1expo 11586 m1exp1 11587 |
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